265 research outputs found
Dyck paths and pattern-avoiding matchings
How many matchings on the vertex set V={1,2,...,2n} avoid a given
configuration of three edges? Chen, Deng and Du have shown that the number of
matchings that avoid three nesting edges is equal to the number of matchings
avoiding three pairwise crossing edges. In this paper, we consider other
forbidden configurations of size three. We present a bijection between
matchings avoiding three crossing edges and matchings avoiding an edge nested
below two crossing edges. This bijection uses non-crossing pairs of Dyck paths
of length 2n as an intermediate step.
Apart from that, we give a bijection that maps matchings avoiding two nested
edges crossed by a third edge onto the matchings avoiding all configurations
from an infinite family, which contains the configuration consisting of three
crossing edges. We use this bijection to show that for matchings of size n>3,
it is easier to avoid three crossing edges than to avoid two nested edges
crossed by a third edge.
In this updated version of this paper, we add new references to papers that
have obtained analogous results in a different context.Comment: 18 pages, 4 figures, important references adde
Anick-type resolutions and consecutive pattern avoidance
For permutations avoiding consecutive patterns from a given set, we present a
combinatorial formula for the multiplicative inverse of the corresponding
exponential generating function. The formula comes from homological algebra
considerations in the same sense as the corresponding inversion formula for
avoiding word patterns comes from the well known Anick's resolution.Comment: 16 pages. Preliminary version, comments are welcom
Operators of equivalent sorting power and related Wilf-equivalences
We study sorting operators on permutations that are obtained
composing Knuth's stack sorting operator and the reversal operator
, as many times as desired. For any such operator , we
provide a size-preserving bijection between the set of permutations sorted by
and the set of those sorted by , proving that these sets are enumerated by the
same sequence, but also that many classical permutation statistics are
equidistributed across these two sets. The description of this family of
bijections is based on a bijection between the set of permutations avoiding the
pattern and the set of those avoiding which preserves many
permutation statistics. We also present other properties of this bijection, in
particular for finding pairs of Wilf-equivalent permutation classes.Comment: 18 pages, 8 figure
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